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基于ADMM正则化轨道算法的高维稀疏精度矩阵估计

王冠鹏,田万,胡涛   

  1. 首都师范大学数学科学学院,北京  100048
  • 出版日期:2021-02-25 发布日期:2021-04-19

王冠鹏,田万,胡涛. 基于ADMM正则化轨道算法的高维稀疏精度矩阵估计[J]. 系统科学与数学, 2021, 41(2): 557-565.

WANG Guanpeng, TIAN Wan,HU Tao. ADMM Algorithmic Regularization Paths for High-Dimensional\\ Sparse Precision Matrix Estimation[J]. Journal of Systems Science and Mathematical Sciences, 2021, 41(2): 557-565.

ADMM Algorithmic Regularization Paths for High-Dimensional\\ Sparse Precision Matrix Estimation

WANG Guanpeng, TIAN Wan ,HU Tao   

  1. School of Mathematics Science, Capital Normal University, Beijing 100048
  • Online:2021-02-25 Published:2021-04-19

考虑预测变量~$p$ 的数量超过样本大小~$n$ 的高维稀疏精度矩阵. 近年来, 由于高维稀疏精度矩阵估计变得越来越流行, 所以文章专注于计算正则化路径, 或者在整个正则化参数范围内解决优化问题. 首先使用定义在正定性约束下最小化Lasso 目标函数精度矩阵估计器, 然后对稀疏精度矩 阵使用乘数交替方向法(ADMM)算法正则化路径, 以快速估计与正则化路径相关联的稀疏模型的序列, 从而进行统计模型 选择. 数值结果表明, 该方法能够快速勾勒出稀疏模型的序列, 不仅克服了计算时间的问题且易于实现, 并且可以在较高分辨率下探索模型空间.

This paper considers high-dimensional sparse precision matrix in which the number of predictors $p$ exceeds the sample size $n$. High-dimensional sparse precision matrix estimation has become more and more popular in the recently years, but we focus on computing regularization paths, or solving the optimization problem over the full range of regularization parameters. We first benefit from a precision matrix estimator which is defined as the minimizer of the Lasso under a positive-definiteness constraint. Then we aim to use the alternating direction method of multipliers (ADMM) algorithmic regularization path for sparse precision matrix to quickly approximate the sequence of sparse models associated with regularization paths for the purposes of statistical model selection. Numerical results show that our method can quickly outline the sequence of sparse models, and this approach not only overcomes the computing time issue, but also easies implementation and explores the model space at a fine resolution.

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